How do I stop image textures from disapearing
First thing that pops to mind is that you do not have an active object using your texture. In the screenshot above, you can see the F button is pressed, so the button itself is dark grey in color. This means that it will be stored regardless of active users. The F stands for Fake, and what it does is simply adding a invisible object that you will never ever see, that has your texture applied to it. As far as I've come to understand it. So make sure this is pressed.

If your texture is still being removed, consider packing it into your .blend file like so: 

You will notice the folder icon next to the + changes, meaning it's packed. This way, it is saved into your .blend file the same way objects, materials, etc are.

Your texture(s) are most likely being removed because Blender deletes unused files upon closing Blender to clear up space. When you make a new material, but decide that you no longer want it, you press the X button to remove it. This does not delete the material. It only gets deleted when you close Blender.

Hope this helps.